Cluster Expansion for Solid Orthohydrogen

The cluster variation method previously applied to calculate the thermal behavior of ferroand antiferromagnets near the Curie temperature is extended to a discussion of solid hydrogen near its order-disorder transition. An attempt is made to describe the relationship between the order-disorder transition and the crystallographic phase change. The orientational free energies of the orthohydrogen or paradeuterium systems are calculated for both the face-centered cubic (fcc) and the hexagonal close-packed (hcp) lattices. The space group $\mathrm{Pa}3\mathrm{}$ is assumed for the fcc case and $\mathrm{Pca}{2}_1$ for the hcp case. Within the cluster variation approximation, a first-order transition is found to occur when the free energies for the fcc and hcp phases cross. The fcc-hcp crystallographic transition occurs near the onset of the orientational ordering in the cubic phase. This prediction is in good agreement with experiment except when repeated cycling is performed.